When does \(SC(X)=\mathbb R^X\) hold?
A map \(f:X\rightarrow Y\) between topological spaces is called scatteredly continuous if for each non-empty subspace \(A\subset X\) the restriction \(f|_A\) has a point of continuity. Let \(SC(X)\) denote the set of all scatteredly continuous maps from \(X\) to the space of real numbers \(\mathbb R\). The authors consider the problem: what conditions on \(X\) imply the equality \(SC(X)=\mathbb R^X\)? The main results: (1)~If \(SC(X)=\mathbb R^X\) then every countable subspace of \(X\) is scattered. (2)~If \(X\) has countable tightness, then \(SC(X)=\mathbb R^X\) iff \(X\) is scattered. (3)~Assume \(V=L\). Then \(SC(X)=\mathbb R^X\) iff \(X\) is scattered. (4)~Assume that \(ZFC\) and \textit{there exists a measurable cardinal} is consistent. Then there exists a completely regular \(0\)-dimensional crowded space \(X\) such that \(SC(X)=\mathbb R^X\).
scientific article; zbMATH DE number 5706259